Year: 2021
Author: Lei-Hong Zhang, Rui Chang
CSIAM Transactions on Applied Mathematics, Vol. 2 (2021), Iss. 2 : pp. 313–335
Abstract
Recent applications in the data science and wireless communications give rise to a particular Rayleigh-quotient maximization, namely, maximizing the sum-of-Rayleigh-quotients over a sphere constraint. Previously, it is shown that maximizing the sum of two Rayleigh quotients is related with a certain eigenvector-dependent nonlinear eigenvalue problem (NEPv), and any global maximizer must be an eigenvector associated with the largest eigenvalue of this NEPv. Based on such a principle for the global maximizer, the self-consistent field (SCF) iteration turns out to be an efficient numerical method. However, generalization of sum of two Rayleigh-quotients to the sum of an arbitrary number of Rayleigh-quotients maximization is not a trivial task. In this paper, we shall develop a new treatment based on the S-Lemma. The new argument, on one hand, handles the sum of two and three Rayleigh-quotients maximizations in a simple way, and also deals with certain general cases, on the other hand. Our result gives a characterization for the solution of this sum-of-Rayleigh-quotients maximization and provides theoretical foundation for an associated SCF iteration. Preliminary numerical results are reported to demonstrate the performance of the SCF iteration.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.2021.nla.04
CSIAM Transactions on Applied Mathematics, Vol. 2 (2021), Iss. 2 : pp. 313–335
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Eigenvector-dependent nonlinear eigenvalue problem self-consistent-field iteration Rayleigh quotient maximization S-Lemma.